To identify the high or low point of the parabola defined by the quadratic equation \( y = -6x^2 + 3x + 8 \), we should first determine the vertex of the parabola. The vertex will either be a high point (maximum) or a low point (minimum) depending on the coefficient of \( x^2 \). Since the coefficient is negative (-6), the parabola opens downward, which means the vertex will be the high point.
The x-coordinate of the vertex for a quadratic equation in the form \( y = ax^2 + bx + c \) can be calculated using the formula:
\[ x = -\frac{b}{2a} \]
In this case:
- \( a = -6 \)
- \( b = 3 \)
Plugging in the values:
\[ x = -\frac{3}{2 \cdot -6} = -\frac{3}{-12} = \frac{1}{4} = 0.25 \]
Now, let's find the y-coordinate by substituting \( x = 0.25 \) back into the original equation:
\[ y = -6(0.25)^2 + 3(0.25) + 8 \]
Calculating \( (0.25)^2 \):
\[ (0.25)^2 = 0.0625 \]
Now substituting into the equation:
\[ y = -6(0.0625) + 3(0.25) + 8 \] \[ y = -0.375 + 0.75 + 8 \] \[ y = 0.375 + 8 \] \[ y = 8.375 \]
So the vertex is at the point \( (0.25, 8.375) \).
Now we can fill in the blanks in the given table. Given the values already known and calculated, we can conclude that:
\[ (0.25, 8.375) \text{ is the high point on the parabola.} \]
Therefore, out of the options provided, the correct response is:
(0.25, 8.375) is the high point on the parabola.
1 answer